Why is squaring a number faster than multiplying two random numbers?
There exist algorithms more efficient than O(N^2) to multiply two numbers (see Karatsuba, Pollard, Schönhage–Strassen, etc.)
The two problems "multiply two arbitrary N-bit numbers" and "Square an arbitrary N-bit number" have the same complexity.
We have
4*x*y = (x+y)^2 - (x-y)^2
So if squaring N-bit integers takes O(f(N)) time, then the product of two arbitrary N-bit integers can be obtained in O(f(N)) too. (that is 2x N-bit sums, 2x N-bit squares, 1x 2N-bit sum, and 1x 2N-bit shift)
And obviously we have
x^2 = x * x
So if multiplying two N-bit integers takes O(f(N)), then squaring a N-bit integer can be done in O(f(N)).
Any algorithm computing the product (resp the square) provides an algorithm to compute the square (resp the product) with the same asymptotic cost.
As noted in other answers, the algorithms used for fast multiplication can be simplified in the case of squaring. The gain will be on the constant in front of the f(N), and not on f(N) itself.
Squaring an n digit number may be faster than multiplying two random n digit numbers. Googling I found this article. It is about arbitrary precision arithmetic but it may be relevant to what your asking. In it the authors say this:
In squaring a large integer, i.e. X^2 = (xn-1, xn-2, ... , x1, x0)^2 many cross-product terms of the form xi * xj and xj * xi are equivalent. They need to be computed only once and then left shifted in order to be doubled. An n-digit squaring operation is performed using only (n^2 + n)/2 single-precision multiplications.